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We investigate error orders for integral limit approximations to traces ofproducts of Toeplitz matrices generated by integrable functions on $[-\pi,\pi]$having some singularities at the origin. Even though a sharp error order of theabove approximation is derived in Theorem~2 of \cite{Lieberman-Phillips-2004},its proof contains an inaccuracy as pointed out by\cite{Ginovyan-Sahakyan-2013}. In the present paper, we reinvestigate the claimgiven in Theorem~2 of \cite{Lieberman-Phillips-2004} and give an alternativeproof of their claim.

Note on Error Bound for Trace Approximation of Products of Toeplitz Matrices